Problem
One interior angle of a regular polygon
A regular dodecagon has twelve equal sides. Its twelve interior angles are all equal too. One of them is labelled x. Find the size of x.
Reasoning · Grade 4-2 Tessellation
Your answer
Watch out The 30 degrees at the centre tempts, but x is two base angles together.
How to solve
Strategy Draw a Diagram — A twelve-sided shape is far too big to attack directly, but a triangle is not. So add lines to the drawing: mark the centre of the dodecagon and join it to all twelve vertices. That slices the whole polygon into twelve identical triangles, and each interior angle of the polygon is simply built from two of the triangles' base angles. The twelve equal angles at the centre must share one full turn, which pins the triangles down completely — a hard polygon problem becomes an easy triangle problem.
1STEP 1
Join the centre to every vertex
Joining the centre makes 12 identical triangles.
12 identical isosceles triangles
A regular shape looks the same from every vertex, so the twelve slices you cut from the centre must all be the same — no calculation is needed to see that.
4.G.A.2Draw A Diagram2STEP 2
Share the full turn at the centre among the twelve triangles
Each centre angle is 360 ÷ 12 = 30 degrees.
360° ÷ 12 = 30°
Sharing one full turn equally among twelve slices is exactly the same kind of division as sharing 360 sweets among 12 children.
4.NBT.B.6Solve An Easier Related ProblemSharing the full turn at the centre among the twelve triangles gives each one the same central angle.
Why?
The twelve central angles fill one complete turn around the centre with no gap and no overlap.
🧱Angles around a point make 360 degreesAngles that fit side by side all the way around one point fill exactly one full turn, 360 degrees.
Why?
Because the polygon is regular the twelve triangles are identical, so the turn is cut into twelve equal shares.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
3STEP 3
Find the two base angles of one triangle
Each base angle is 75 degrees.
(180° - 30°) ÷ 2 = 150° ÷ 2 = 75°
The triangle angle sum plus the fact that equal sides face equal angles is enough to finish any isosceles triangle once one of its angles is known.
8.G.A.5Solve An Easier Related Problem4STEP 4
Rebuild one interior angle from two base angles
Two base angles together make x 150 degrees.
x = 75° + 75° = 150°
An angle cut in two by a line through its corner is just the two pieces added back together — angle measure adds up like lengths do.
4.MD.C.7Draw A Diagram5STEP 5
Check with the general rule for regular polygons
The general rule also gives 150 degrees.
180° × (12 - 2) ÷ 12 = 1800° ÷ 12 = 150°
Testing the shapes you already know — triangle, square, hexagon — is a quick way to be sure a general rule has been remembered correctly.
8.G.A.5Look For A PatternAnswer
150 degrees
180 × 10 ÷ 12 = 150
150° is obtuse but well short of a straight 180°, which is exactly how the corners of the drawing look — a dodecagon is nearly round, so each corner is only a gentle bend. A second check: at each vertex the interior angle and the exterior turn together make a straight line, so the turn is 180° - 150° = 30°; walking right around the outline means twelve such turns, 12 × 30° = 360°, one complete turn back to the starting direction, as it must be. A third check: 12 × 150° = 1800°, which matches the interior-angle total 180° × (12 - 2) = 1800° for any twelve-sided polygon.
Takeaway
Cut a regular polygon into slices from its centre — one full turn shared out tells you everything about the corners.
- Join the centre to every vertex
- Share the full turn at the centre among the twelve triangles
- Find the two base angles of one triangle
- Rebuild one interior angle from two base angles
- Check with the general rule for regular polygons